Zero-range Distorted-wave Born Approximation Research Articles

Model calculation of the differential cross sections and angle-integrated cross sections of the emitted triton for neutron-induced Li6 reactions at low incident energies

Regarded as important for applications and theoretical studies, the differential cross sections of the emitted triton for the neutron-induced $^{6}\mathrm{Li}$ reaction at low incident energies were calculated, based on the zero-range distorted-wave Born approximation theory with the assumption of $^{6}\mathrm{Li}$ nucleus consisting of $t{+}^{3}\mathrm{He}$ or $d+\ensuremath{\alpha}$. As a function of widths and excited energies of the discrete energy levels, an effective excited energy formula was proposed to describe their partial contributions. In addition, the optical model potential parameters, which had been successfully used to reproduce the double-differential cross sections of the emitted neutrons in our previous works in incident energy range from 5.0 to 20.0 MeV were extended in a low-energy range from 1.0 eV to 3.0 MeV in this paper. The calculated results agreed well with the measured differential cross sections recently published in 2020 and were further consistent with the measured angle-integrated cross sections. This indicates that the knock-out process and heavy-particle knockout process were dominant in an energy range from 1.0 eV to 3.0 MeV, whereas the shapes of the measured angular distributions in incident energy range from 0.1 to 1.0 MeV could be successfully explained by the Hauser-Feshbach model.

Study of 26Mg through 1p pick-up reaction 27Al(d,3He)

The reaction [Formula: see text]Al([Formula: see text],3He) at 25 MeV beam energy has been utilized to study the states in even–even nucleus [Formula: see text]Mg. The spectroscopic factors have been extracted for the states of [Formula: see text]Mg up to 7.50[Formula: see text]MeV excitation energy using local, zero-range distorted wave Born approximation. The extracted spectroscopic factors have been compared with the previously reported values. The present results were also compared with the predictions from a theoretical shell model and rotational model.

Incident-energy dependence of the analyzing power in the58Ni(p,3He)56Co reaction between 80 and 120 MeV

This project looks at the angular distributions of the differential cross sections and analyzing powers of a few low lying states of 56 Co in the reaction 58 Ni(p, 3 He) 56 Co at three different incident energies between 80 and 120 MeV. The measurements are compared with zero-range distorted-wave Born approximation (DWBA) calculations in which we assume a simple direct two-nucleon pickup process. Earlier inclusive (p, 3 He) reaction studies on similar targets were successfully treated in terms of a statistical pre-equilibrium multistep formalism, in which the final stage of the reaction involved a deuteron pickup, described by means of the DWBA. The analyzing power was shown to be rather sensitive to the contributions of the different order steps. However some features observed in the analyzing powers of these inclusive studies, though reproduced by the theory, are not fully understood. We therefore investigate the ability of the DWBA model to describe the (p, 3 He) pickup reaction to discrete states at different incident energies using a high resolution spectrometer.

Probing the pairing interaction through two-neutron transfer reactions

The treatment of the pairing interaction in mean-field-based models is addressed. In particular, the possibility to use pair transfers as a tool to better constrain this interaction is discussed. First, pairing inter- actions with various density dependencies (surface/volume mixing) are used in the microscopic Hartree-Fock- Bogoliubov + quasiparticle random-phase approximation model to generate the form factors to be used in reac- tion calculations. Cross sections for (p,t) two-neutron transfer reactions are calculated in the one-step zero-range distorted-wave Born approximation for some Tin isotopes and for incident proton energies from 15 to 35 MeV. Three different surface/volume mixings of a zero-range density-dependent pairing interaction are employed in the microscopic calculations and the sensitivity of the cross sections to the different mixings is analyzed. Differences among the three different theoretical predictions are found espacially for the nucleus 136 Sn and they are more important at the incident proton energy of 15 MeV. We thus indicate (p,t) two-neutron transfer reactions with very neutron-rich Sn isotopes and at proton energies around 15 MeV as good experimental cases where the sur- face/volume mixing of the pairing interaction may be probed. In the second part of the manuscript, ground-state to ground-state transitions are investigated. Approximations made to estimate two-nucleon transfer probabilities in ground-state to ground-state transitions and the physical interpretation of these probabilities are discussed. Probabilities are often calculated by approximating both ground states of the initial nucleus A and of the final nucleus A ±2 by the same quasiparticle vacuum. We analyze two improvements of this approach. First, the effect of using two different ground states with average numbers of particles A and A ±2 is quantified. Second, by using projection techniques, the role of particle number restoration is analyzed. Our analysis shows that the improved treatment plays a role close to magicity, leading to an enhancement of the pair-transfer probability. In midshell regions, part of the error made by approximating the initial and final ground states by a single vacuum is com- pensated by projecting onto a good particle number. Surface effects are analyzed by using pairing interactions with a different volume/surface mixing. Finally, a simple expression of the pair-transfer probability is given in terms of occupation probabilities in the canonical basis. We show that, in the canonical basis formulation, surface effects that are visible in the transfer probability are related to the fragmentation of single-particle occupancies close to the Fermi energy. This provides a complementary interpretation with respect to the standard quasiparti- cle representation where surface effects are generated by the integrated radial profiles of the contributing wave functions.

Probing the pairing interaction through two-neutron transfer reactions

Cross sections for ($p,t$) two-neutron transfer reactions are calculated in the one-step zero-range distorted-wave Born approximation for the tin isotopes $^{124}$Sn and $^{136}$Sn and for incident proton energies from 15 to 35 MeV. Microscopic quasiparticle random-phase approximation form factors are provided for the reaction calculation and phenomenological optical potentials are used in both the entrance and the exit channels. Three different surface/volume mixings of a zero-range density-dependent pairing interaction are employed in the microscopic calculations and the sensitivity of the cross sections to the different mixings is analyzed. Since absolute cross sections cannot be obtained within our model, we compare the positions of the diffraction minima and the shapes of the angular distributions. No differences are found in the position of the diffraction minima for the reaction $^{124}$Sn($p,t$)$^{122}$Sn. On the other side, the angular distributions obtained for the reaction $^{136}$Sn($p,t$)$^{134}$Sn with surface and mixed interactions differ at large angles for some values of the incident proton energy. For this reaction, we compare the ratios of the cross sections associated to the ground state and the first excited state transitions. Differences among the three different theoretical predictions are found and they are more important at the incident proton energy of 15 MeV. As a conclusion, we indicate ($p,t$) two-neutron transfer reactions with very neutron-rich Sn isotopes and at proton energies around 15 MeV as good experimental cases where the surface/volume mixing of the pairing interaction may be probed.

Resonance effects in distorted-wave Born approximation analyses of(d→,p)analyzing powers at very low energies

In very low energy $(d,p)$ reactions, resonance effects are explicitly taken into account for zero-range distorted-wave Born approximation calculations by restricting the reaction amplitudes to those specified by the spin-parity of the resonance. The application of formulas of tensor analyzing powers to $^{6}\mathrm{Li}(\stackrel{\ensuremath{\rightarrow}}{d},p)^{7}\mathrm{Li}$(g.s.) reactions in resonance regions provides us a clear understanding of the characteristics of the measured analyzing powers and aides in the determination of the spin-parity of the resonance states concerned at ${E}_{d}=90$, $600$, and $960$ keV.

High spin two-neutron-hole states via the208Pb(p,t)206Pbreaction at 120 MeV

Cross sections and analyzing powers were obtained for the reaction ${}^{208}\mathrm{Pb}(p,t){ }^{206}\mathrm{Pb}$ at ${T}_{p}=120\mathrm{MeV},$ up to an excitation energy of 6.5 MeV, and with an energy resolution of 25--40 keV. Momentum matching considerations strongly favor the population of two-neutron-hole states with $J>~6.$ Making use of cross section and analyzing power angular distributions, 14 natural parity states were identified with $J=6$ to 12, including all of the states with $J>~9$ expected in the $N=5$ shell model space. The data were analyzed using the local zero-range distorted-wave Born approximation. Theory was normalized to experiment by making use of the sum rules for two-nucleon transfer reactions. Enhancement factors (ratio of experiment to normalized theory) were obtained, both for single, pure-configuration, and mixed configuration two-particle amplitudes from the shell model calculations of Kuo and Herling. Some of the details of the observed distribution of strengths were better accounted for by the shell model calculations, but, overall, the pure-configuration assumption accounted for the enhancement factors and energies about as well.

0f7/2 proton transfer on 55Mn.

Proton pickup data have been obtained through the {sup 55}Mn({ital d},{sup 3}He){sup 54}Cr ({ital E}{sub {ital d}} = 46 MeV) reaction in the 6{degree}{endash}30{degree} angular range using a magnetic spectrometer with a resolution of {approximately} 40 keV full width at half maximum. Spectroscopic factors associated with transitions to 24 {sup 54}Cr final states ({ital E}{sup {asterisk}}{le} 6.104 MeV) were determined from local, zero-range distorted-wave Born approximation analyses of the measured angular distributions allowing for l = 0, 1, 2, and 3 transfer. A spin-dependent sum-rule analysis of the 0{ital f}{sub 7/2} proton transfer data has been performed using complementary stripping data from a study of the {sup 55}Mn({alpha},{ital t}){sup 56}Fe reaction. The 0{ital f}{sub 7/2} proton transfer data have also been compared to results from a 0{ital f}1{ital p} shell-model calculation based on a new effective interaction for {ital A} = 41{endash}66 nuclei. {copyright} {ital 1996 The American Physical Society.}

Shell-model study of 40Ca with the 56-MeV (d

Single-particle states in [sup 41]Ca have been investigated through the Ca([ital [rvec d]],[ital p])[sup 41]Ca reaction with polarized deuterons of [ital E][sub [ital d]]=56 MeV. Angular distributions of differential cross sections and vector analyzing powers were measured for 91 transitions up to 9.7-MeV excitation energy by using a magnetic spectrograph. Analyses of these data were performed to determine [ital j][sup [pi]] and [ital C][sup 2][ital S] of single-particle levels with zero-range distorted-wave Born approximation calculations using the adiabatic approximation. The occupation probabilities and the single-particle energies in [sup 40]Ca were extracted for the valence (2[ital p][minus]1[ital f]) and (2[ital s][minus]1[ital d]) shells with including previous ([ital p],[ital d]) data. The results on these quantities show good agreement with theoretical expectations.

Cluster structures in 10Be from the 7Li( alpha,p)10Be reaction.

Three-nucleon transfer by the $^{7}\mathrm{Li}$(\ensuremath{\alpha},p${)}^{10}$Be reaction at 65 MeV has been carried out with good resolution. A cluster model is compared to the data with the aid of zero-range distorted wave Born approximation reaction calculations, finding consistent spectroscopic factors for low-lying K=${0}^{+}$ and ${1}^{\mathrm{\ensuremath{-}}}$ bands. Large cluster spectroscopic factors are found for the states at 17.74 and 18.54 MeV that serve as resonances for reactions between $^{3}\mathrm{H}$ and $^{7}\mathrm{Li}$ at low energies of astrophysical importance. A research for states very near threshold found no transitions that significantly alter the important astrophysical reaction rate.

Stretched high-spin two-neutron-hole states in 206Pb and 114Sn via the (p,t) reaction at 168 MeV.

Two-neutron-hole excitations have been investigated via the(/ital p/,/ital t/) reaction at 168 MeV on /sup 208/Pb and/sup 116/Sn targets, up to /similar to/25 and /similar to/17 MeVexcitation energy, respectively. The kinematic conditions strongly favor high/ital L/ transfer values (/ital L//similar to/10 for Sn and /ital L//similar to/13 for Pb).High-spin stretched states are found to dominate the spectra with an evenstronger enhancement for/ital J/=((/ital l//sub 1/+(1/2)(l/sub 2//minus/1)/2))/sub /ital J//sub max// states.Angular distributions and standardlocal zero-range distorted-wave Born approximation analysis have been performedfor /sup 206/Pb. Such analysis has been extended to the/sup 114/Sn data taken at two angles. The zero-range normalizationconstant /ital D//sub 0//sup 2/ is found to be significantly smaller thanthat usually adopted at low incident energy, /ital D//sub 0//sup 2//similar to/6instead of /similar to/22. The high-spin stretched states of the valencemultiplets are distributed from 2.2 MeV up to 6.1 MeV in/sup 206/Pb, whereas they are clustered between /similar to/3.0and 3.7 MeV in /sup 114/Sn. New spectroscopic information gainedfor /ital J//sup ..pi../greater than or equal to6 levels or structures in /sup 206/Pb and/sup 114/Sn is discussed together with that of previous studies,especially the (..cap alpha..,/sup 6/He) results at high incident energy.Such comparisons show that configuration mixing is larger than predicted forthemore » higher-energy transitions in /sup 206/Pb. No significantconcentration of valence plus deep hole states could be found in/sup 206/Pb, whereas the previously known one valence plus one deepbump in /sup 114/Sn exhibits a maximum at /similar to/7.3 MeVhere mainly attributed to the(/ital g//sub 7/2/+g/sub 9/2/)/sub 8/sup +// state inagreement with the prediction of the quasiparticle phonon model.« less

Neutron-hole strengths and core coupling in 87Sr via the 88Sr(3He, alpha )87Sr reaction.

Neutron-hole states in /sup 87/Sr were studied by means of the /sup 88/Sr(/sup 3/He,..cap alpha..)/sup 87/Sr reaction at 36 MeV. Angular distribution measurements were carried out from 3/sup 0/ to 41/sup 0/ (lab) and analyzed with the zero-range distorted-wave Born approximation method. Spectroscopic factors have been determined for about 50 discrete levels in /sup 87/Sr located below 6 MeV excitation energy and for the three lowest isobaric analog states 2p(3/2, 1f(5/2, and 2p(1/2 observed around 11 MeV. Many l = 1 and l = 3 discrete levels are observed in the 3--6 MeV excitation energy range. In addition, a large part of the 1f-2p strength is found to lie in the higher-lying continuum up to 13 MeV (about 10% and 40% for the l = 1 and 3 contributions, respectively). The distribution of the 1f-2p neutron-hole strength is compared to previous data on neighboring nuclei /sup 89/Zr and /sup 91/Mo. In addition, angular distributions for some low-lying high-spin states in /sup 87/Sr are nicely reproduced by coupled-channel calculations which assume pure two-step (inelastic plus g(9/2 transfer) processes. Spin-parity assignments are proposed for the levels at 1.92 MeV ((7/2/sup +/), 2.16 MeV ((11/2/sup +/), 2.55 MeV ((9/2/sup -/), and 2.60 MeVmore » ((13/2/sup -/), on the basis of this coupled-channel analysis.« less

Nuclear structure of 86,87Rb from 87,88Si(d, 3He) at 28 MeV

Differential cross sections for 87Sr(d, 3He) 86Rb and 88Sr(d, 3He) 87Rb were measured with the Princeton 27.9 MeV deuteron beam and the quadrupole-dipole-dipole-dipole (QDDD) spectrograph. States of 86Rb and 87Rb up to 1.74 MeV and 1.58 MeV, respectively, were studied with a total resolution of 16keV. Zero-range distorted-wave Born approximation (DWBA) calculations were used to determine l transfers and spectroscopic strengths. If the targets are described as a closed shell for 88Sr and as (v g 9 2 ) −1 for 87Sr, only the 88Sr(d, 3He) 87Rb spectroscopic strengths are close to shell model expectations. The observed 87Sr(d, 3He) 86Rb strengths fall 25% below the shell model sum rule. Hence the data suggest that the proton wave functions of 87Sr contain appreciably more configuration mixing than those of 88Sr. Values for the (πf 5 2 , v g 9 2 ) residual interaction matrix elements are proposed, but must remain tentative until the shortfall of f 5 2 strength is understood.

Location of multiparticle-multihole strength in 16N via the 3-particle transfer reaction 13C(α, p) 16N at 118 MeV

Differential cross sections were measured for the 13C(α, p) 16N reaction at E α = 118 MeV for an excitation energy range up to 14.5 MeV. Zero-range distorted wave Born approximation (DWBA) calculations were performed using microscopic form factors. Spin-parity assignments are suggested for states at 11.21 MeV (6 −) and 11.81 MeV (7 −) on the basis of Bansal-French-Zamick weak coupling calculations and DWBA calculations. Arguments from 16O(e, e'), 16O(p, p') and the present experiment are given relating to the location of J π = 4 −, T = 1 strength in 16N.

Study on the 208Pb(α, 6He) 206Pb reaction at Eα = 109 MeV

The 208Pb(α, 6He) 206Pb reaction induced by 109 MeV α particles has been studied with 70 keV energy resolution. Twenty-five transitions up to 6.2 MeV in excitation energy have been analyzed. The shapes of the angular distributions are reproduced well by the zero-range distorted-wave Born approximation (DWBA) calculations. The spin and parity of the level at 6.10 MeV are proposed to be 9 −. It is interpreted as a weakly-coupled state of the collective 3 − state and the two-hole 6 + shell-model state. The energy dependence of the total cross sections of the (α, 6He) and (p, t) reactions has been studied systematically by using our present data and other data from the literatures. It reveals that there exists an empirical rule which allows one to compare quantitatively the ratios of the transition strengths of both reactions. The transition strength of the (α, 6He) reaction is found to be about 35 times smaller than that of the (p, t) reaction for the transitions to the natural parity states, while it is about 15 times smaller for the transition to the unnatural parity 3 + state. These small transition strengths are explained by the fact that this reaction occurs at the surface region of the nucleus.

17O(p

Differential cross sections and analyzing powers of the /sup 17/O(p,t) reaction were measured over the angular range from 5/sup 0/ to 55/sup 0/ at T/sub p/ = 89.7 MeV. The data for transitions to five residual states of /sup 15/O are compared with calculations in the finite-range and zero-range distorted-wave Born approximation. The distorting potential parameters were obtained from optical-model analyses of elastic proton and helion scattering. The (p,t) cross sections are underpredicted by the finite-range calculations and overpredicted by zero-range calculations based on standard normalizations. The reproduction of the analyzing powers is only fair. Two-step calculations of the type (p,d;d,t) were performed in a zero-range, second-order distorted-wave Born approximation. The ground state and six one-particle, one-hole states of /sup 16/O were taken into account as intermediate states. Consistent improvement in the prediction of the data was not obtained.

Resonating group method as applied to the spectroscopy ofα-transfer reactions

In the conventional approach to $\ensuremath{\alpha}$-transfer reactions the finite- and/or zero-range distorted-wave Born approximation is used in liaison with a macroscopic description of the captured $\ensuremath{\alpha}$ particle in the residual nucleus. Here the specific example of $^{16}\mathrm{O}$($^{6}\mathrm{Li}$,d)$^{20}\mathrm{Ne}$ reactions at different projectile energies is taken to present a microscopic resonating group method analysis of the $\ensuremath{\alpha}$ particle in the final nucleus (for the reaction part the simple zero-range distorted-wave Born approximation is employed). In the discussion of suitable nucleon-nucleon interactions, force number one of the effective interactions presented by Volkov is shown to be most appropriate for the system considered. Application of the continuous analog of Newton's method to the evaluation of the resonating group method equations yields an increased accuracy with respect to traditional methods. The resonating group method description induces only minor changes in the structures of the angular distributions, but it does serve its purpose in yielding reliable and consistent spectroscopic information.NUCLEAR STRUCTURE $^{16}\mathrm{O}$($^{6}\mathrm{Li}$,d)$^{20}\mathrm{Ne}$; $E=20 \mathrm{to} 32$ MeV; calculated $B(E2)$; reduced widths, $\frac{d\ensuremath{\sigma}}{d\ensuremath{\Omega}}$; extracted $\ensuremath{\alpha}$-spectroscopic factors. ZRDWBA with microscope RGM description of residual $\ensuremath{\alpha}$ particle in $^{20}\mathrm{Ne}$; application of continuous analog of Newton's method; tested and applied Volkov force No. 1; direct mechanism.

Energy dependence of the zero-range normalization constant for the (p,t) reaction at incident energies of 34.9, 45.1, 54.7, and 65.0 MeV

Angular distributions of $^{124}\mathrm{Sn}$($p$,$t$) reactions to low-lying states of $^{122}\mathrm{Sn}$ were measured at ${E}_{p}=34.9, 45.1, 54.7, \mathrm{and} 65.0$ MeV and analyzed with zero-range distorted-wave Born approximation theory. It is found that the zero-range normalization constant ${D}_{0}^{2}$ is smoothly decreasing as a function of the incident proton energy in the several tens of MeV region of bombarding energy.NUCLEAR REACTIONS $^{124}\mathrm{Sn}$($p$,$t$), $E=34.9, 45.1, 54.7, \mathrm{and} 65.0$ MeV; measured (${E}_{t}$, ${\ensuremath{\theta}}_{t}$); DWBA analysis; deduced zero-range normalization factor.

Microscopic form factor for DWBA analysis of light-ion induced three-nucleon transfer reactions

A modified zero-range distorted-wave Born approximation calculation was performed for the (p, α) reactions. The form factor is calculated by a first method, which is an extension of the Bayman and Kallio method to extract the relative l = 0 part of two-nucleon shell-model wave functions. The method includes some of the range effects so that one can predict the absolute magnitudes of the cross sections. A second method in finite range involves the 3He cluster expansion of the α-particle where the known ( 3He, α) transfer normalization may be used to estimate the absolute cross sections. The 118Sn(p, α) 115In reactions are analyzed using the hole-vibration coupling model for 115In. The shapes of the experimental cross sections and the analyzing powers can be fitted by the calculations but the magnitudes of the cross sections are predicted to be too small.

Microscopic analysis of the12C(6Li,d)16O reaction

The12C(6Li,d)16O reaction at 28 MeV and 34 MeV bombarding energy are analized in Zero Range Distorted Wave Born Approximation (ZR-DWBA) utilizing microscopic four-particle transfer form factors. Configuration mixing is taken into account utilizing the Zuker, Buck and McGrory wave functions for16O. Good agreement is found for the angular distributions. For the relative cross sections to the different excited states, acceptable agreement is found for the 1− state (7.12 MeV) and 4+ (10.56 MeV), while the model clearly underestimates the transition to the 2+ state (6.92 MeV).